Irreducible Representations of $C(X,M_{n}(\mathbb{C}))$ as a C$^{*}$-algebra

Question

Suppose $X$ is a compact Hausdorff space. Let $\varphi$ be an irreducible representation of $C(X,M_{n}(\mathbb{C}))$.

Is it true that $\varphi$ must be unitarily equivalent to a point evaluation?

Answer 1

The algebra $A:=C(X, M_n)$ is isomorphic to $C(X)\otimes M_n$. In this case, irreducible representations of $A$ are always coming from the tensor product of irreducible representations of $C(X)$ and $M_n$ (see for example Corollary 3.4.3 in Brown-Ozawa's book).

Irreducible representations of $C(X)$ are unitarily equivalent to point evaluations and any irreducible representation of $M_n$ is unitarily equivalent to the identity (you can also find this in Murphy's book).

Therefore, any irreducible representation of $A$ is unitarily equivalent to a representation of the form $\mathrm{ev}_x \otimes \mathrm{id}_{M_n}$ with $x \in X$.